What are beats ? A tuning fork gives 6 beats per second with 25\cdot 4\text{ cm} length of stretched sonameter wire. The same 6 beats are heard per second when the length of the same wire is reduced to 24\cdot 6\text{ cm}. Calculate the frequency of the tuning fork.
What is achromatism ? Obtain the condition for achromatism of two thin lenses separated by a distance of x metres.
Write an expression for the profile (t = 0) of a harmonic wave moving in the +x-direction such that at x = 0, \psi = 10; at x = \lambda/6, \psi = 20 and at x = 5\lambda/12, \psi = 0. (The notations carry usual meanings)
A glass chamber 25\text{ mm} long filled with air is positioned in front of one of the secondary sources in Young's experiment. The air is removed from the chamber and a test gas is entered in its stead. On comparing the fringe system corresponding to air with that of the test gas, it is found that the entire interference pattern on the screen was displaced by 21 bright bands towards gas-containing chamber. Given that \lambda = 656{\cdot}2816\text{ nm} for which the refractive index of air is n_a = 1{\cdot}000276. Determine the refractive index of the gas (n_g).
Consider the formation of Newton's rings when two closely spaced wavelengths are present, for example, the \text{D}_1 and \text{D}_2 lines of sodium (\lambda_1 = 5890\text{ \AA} and \lambda_2 = 5896\text{ \AA}). What will be the effect of the presence of these two wavelengths as the lens is gradually moved away from the plate? What will happen if the sodium lamp is replaced by a white source? (The radius of curvature of the lens, R = 100\text{ cm})
(i) Obtain an expression for the displacement of the damped harmonic oscillator where the damping force is proportional to the velocity. Discuss the effect of the damping on the displacement and frequency of the oscillator. Discuss the difference physically between overdamped and critically damped by taking example of a pendulum. (ii) A particle of mass 3 moves along the x-axis attracted towards the origin by a force whose magnitude is numerically equal to 12x. The particle is also subjected to damping force whose magnitude is numerically equal to 12 times the instantaneous speed. If it is initially at rest at x = 10, find the position and the velocity of the particle at any time.
What do you understand by quarter wave plate? How can it be used to convert a linearly polarized wave into a circularly polarized wave?
(i) Describe the wave given by the expression E = \hat{j} E_0 \sin \left( \frac{2\pi x}{\lambda} - \omega t \right) - \hat{k} E_0 \sin \left( \frac{2\pi x}{\lambda} - \omega t \right) where \hat{j} and \hat{k} are unit basis vectors in Cartesian coordinates. (ii) Write an expression for linearly polarized harmonic plane wave of scalar amplitude E_0, propagating along a line in the xy-plane at 45^\circ to the x-axis and having the xy-plane as its plane of vibration.
A High Frequency (HF) radio receiver receives simultaneously two signals from a transmitter 400\text{ km} away, one by a path along the surface of the Earth, and the other by reflection from a portion of the ionospheric layer situated at a height of 200\text{ km}. We assume that the Earth is flat and the ionospheric layer acts as a perfect horizontal reflector, which is moving slowly in the vertical direction. When the frequency of the transmitted wave is 10\text{ MHz}, it is observed that the combined signal strength varies from maximum to minimum and back to maximum 6 times per minute. With what slow vertical speed is the ionospheric layer moving ?
How can the refractive index of a liquid be determined using a method of Newton's rings ? Obtain an appropriate expression for calculating the refractive index.
The quality factor Q in a damped harmonic motion is defined as Q = \frac{2\pi \times \text{Average energy stored per cycle}}{\text{Average energy dissipated per cycle}} Show that Q = \frac{\omega}{2b} (where the symbols have usual meaning).
Two Fabry -- Pérot interferometers have same plate separation but the coefficients of intensity reflection are 0\cdot 7 and 0\cdot 9. Deduce the relative width of the maxima in the above two cases.
With a neat ray diagram, describe Fraunhofer diffraction at a circular aperture.
Explain Huygens' theory of double refraction in a uniaxial crystal. Discuss the formation of ordinary and extraordinary rays for oblique incidence when an optic axis is in the plane of incidence and when it is perpendicular to the plane of incidence.
It is required that a real image twice the size of the object be formed by a thin plano-convex lens. If the lens has a radius of curvature of 50\text{ cm} and a refractive index of \mu = 1\cdot 5, determine the locations of the object and image with respect to the lens.
In an \text{He}-\text{Ne} laser system, the two energy levels of \text{Ne} involved in lasing action have energy values of 20\cdot 66\text{ eV} and 18\cdot 70\text{ eV} respectively. Population inversion occurs between these two energy levels. What will be the wavelength of the laser beam produced ? What will be the population in the metastable energy level with respect to the upper excited level at room temperature (27^\circ\text{C}) ?
Thin transparent films produce colours when a white light falls on them or they act as antireflecting coatings. Find the conditions for these characteristics.
(i) Give the equation of motions for a particle executing simple harmonic motion for undamped, damped and forced vibrations, and explain the terms.
(ii) Give a plot between the displacement and time for each case of the above and interpret.
(iii) What are the conditions for critical damping and resonance?
(i) When a monochromatic light passes through a single slit, dark and bright fringes are observed on a screen kept far away from the slit. Draw an amplitude and intensity pattern of the light on the screen.
(ii) Write an expression for intensity distribution and give the conditions for bright and dark fringes.
(iii) If the slit width is 0\cdot 3\text{ mm} and the distance between the screen and the slit is about 30\text{ cm}, then what is the angle of diffraction for the first minimum? (\lambda = 5 \times 10^{-5}\text{ cm})
With the help of a neat diagram, explain spherical aberration. Briefly discuss the methods to minimize spherical aberration.